# Integrating Factors (IB Mathematics AA HL) | Maths with Lemon

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### Integrating Factors

IB Mathematics: Analysis and Approaches Higher Level · Lesson 10.4 · AHL 5.18

Learn how to solve first-order linear differential equations by recognising a product-rule derivative.

Lesson 10.4 worksheet (PDF) · Previous lesson: Separable and Homogeneous Differential Equations


#### 1. When Should You Use an Integrating Factor?


##### What you need to know

- The equation must be linear in \(y\) and its first derivative.
- Rearrange it into standard form before identifying \(P(x)\) and \(Q(x)\).
- The integrating factor converts the left side into one product-rule derivative.
- Use any initial condition only after obtaining the general solution.

##### Standard form

The integrating factor is

Multiplying the equation by \(\mu(x)\) gives


#### 2. Why Does the Method Work?


##### The product rule connection

- Recall the product rule for \(\mu(x)y\).
- Choose \(\mu\) so that its derivative is \(P(x)\mu\).
- Then integrate the whole product derivative in one step.

##### Derivation


#### 3. Worked Example from Your Worksheet


##### Solution strategy

- Identify \(P(x)=2/x\).
- Find the integrating factor.
- Multiply through and simplify using the product rule.
- Integrate and isolate \(y\).

##### Solve for \(x>0\)

Substitute into the original equation to check your result.


#### 4. Exponential and Trigonometric Examples


##### Spot the integrating factor

- For \(y^{\prime}+ay=Q(x)\), use \(e^{ax}\).
- For \(P(x)=1/x\), use \(|x|\) on an interval excluding zero; \(x\) is convenient for \(x>0\).
- For \(P(x)=\tan x\), an integrating factor is \(\sec x\) on intervals where \(\cos x\ne0\).

##### Quick worked examples


#### 5. Particular Solutions and Initial Conditions


##### Exam method

- Find the general solution first.
- Substitute the given point \((x_0,y_0)\).
- Solve for the constant \(C\).
- State the interval of validity if the equation has a restricted domain.

##### Worksheet-style example

Using \(y(1)=5\), we obtain \(C=2\).


#### 6. Application: Falling Through a Liquid


##### Interpret the model

- Velocity changes due to a constant driving term and a resistive term.
- A terminal velocity occurs when the acceleration becomes zero.
- An integrating factor gives the full velocity function, not only the terminal value.

##### Worksheet model

As \(t\to\infty\), the exponential term tends to zero.


#### 7. Interactive Integrating Factor Explorer


##### Explore a linear equation

- Change the coefficient \(a\), the forcing \(b\), and the initial value.
- See the exact solution of \(y^{\prime}+ay=b\).
- Observe how the solution approaches the equilibrium \(b/a\) when \(a>0\).

##### Model: \(y^{\prime}+ay=b\)


#### 8. HL Practice and Revision


##### From your Lesson 10.4 worksheet

- Solve \(y^{\prime}-y=e^x\).
- Solve \(y^{\prime}+y/x=x^2\).
- Investigate integrating factors involving \(\tan x\).
- Apply substitutions such as \(z=y^2\) or \(z=\sin y\) to obtain linear equations.
- Work through the extension problems involving falling motion and second-order equations.

##### Common mistakes

- Forgetting to divide by the coefficient of \(y^{\prime}\) before finding \(P(x)\).
- Missing the constant of integration.
- Confusing \(\mu y\) with \(\mu\,dy/dx\).
- Dropping domain restrictions in logarithmic or trigonometric integrating factors.
Open the full worksheet


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